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Applied Mathematics and Mechanics

, Volume 37, Issue 12, pp 1587–1596 | Cite as

Heat transfer characteristics of thin power-law liquid films over horizontal stretching sheet with internal heating and variable thermal coefficient

  • Yanhai LinEmail author
  • Liancun Zheng
  • Lianxi Ma
Article

Abstract

The effect of internal heating source on the film momentum and thermal transport characteristic of thin finite power-law liquids over an accelerating unsteady horizontal stretched interface is studied. Unlike most classical works in this field, a general surface temperature distribution of the liquid film and the generalized Fourier’s law for varying thermal conductivity are taken into consideration. Appropriate similarity transformations are used to convert the strongly nonlinear governing partial differential equations (PDEs) into a boundary value problem with a group of two-point ordinary differential equations (ODEs). The correspondence between the liquid film thickness and the unsteadiness parameter is derived with the BVP4C program in MATLAB. Numerical solutions to the self-similarity ODEs are obtained using the shooting technique combined with a Runge-Kutta iteration program and Newton’s scheme. The effects of the involved physical parameters on the fluid’s horizontal velocity and temperature distribution are presented and discussed.

Key words

non-Newtonian fluid nonlinear equation thin film heat transfer internal heating stretching sheet thermal conductivity numerical solution 

Nomenclature

a, b

positive constant, s-1

Cp

specific heat capacity, J · kg-1 · K-1

d

positive constant, m-r1

f

dimensionless stream function

h

liquid film thickness, m

K

viscosity coefficient, kg · m-1 · sn-2

k

effective thermal conductivity, W· m-1 · K-1

n

power-law index

Pr

generalized Prandtl number

Rex

local Reynolds number

r1

r 2, power indices

S

unsteadiness parameter

S0

critical value

T

temperature, K

T0

temperature at origin, K

Tref

standard temperature, K

Ts

temperature of stretched surface, K

t

time, s

u, v

liquid velocity components along with x-direction and y-direction, respectively, m · s-1

us

horizontal velocity of stretched surface, m · s-1

x, y

streamwise coordinate and cross-stream coordinate, respectively, m.

Greek symbols

β

dimensionless film thickness

η

similarity variable

θ

dimensionless temperature

ρ

density, kg · m-3

τxy

modified shear viscous drag, N · m-2

φ

heating source parameter

ψ

stream function, m2 · s-1

ω

consistency thermal coefficient, kg · m · sn-4 · K-1 s, for wall surface or stretched surface.

Chinese Library Classification

O35 O373 

2010 Mathematics Subject Classification

76A05 76A20 80A20 34B15 

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Copyright information

© Shanghai University and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.School of Mathematical SciencesHuaqiao UniversityFujian ProvinceChina
  2. 2.School of Mathematics and PhysicsUniversity of Science and Technology BeijingBeijingChina
  3. 3.Department of PhysicsBlinn CollegeBryanUSA

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